3.2784 \(\int \frac{(c x)^{-1+\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]

[Out]

(2*(c*x)^(n/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(Sqrt[b]*c*n*x^(n/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0300451, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {347, 345, 217, 206} \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 + n/2)/Sqrt[a + b*x^n],x]

[Out]

(2*(c*x)^(n/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(Sqrt[b]*c*n*x^(n/2))

Rule 347

Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracP
art[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !In
tegerQ[n]

Rule 345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(c x)^{-1+\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx &=\frac{\left (x^{-n/2} (c x)^{n/2}\right ) \int \frac{x^{-1+\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx}{c}\\ &=\frac{\left (2 x^{-n/2} (c x)^{n/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^{n/2}\right )}{c n}\\ &=\frac{\left (2 x^{-n/2} (c x)^{n/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{n/2}}{\sqrt{a+b x^n}}\right )}{c n}\\ &=\frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n}\\ \end{align*}

Mathematica [A]  time = 0.0245547, size = 78, normalized size = 1.44 \[ \frac{2 \sqrt{a} x^{-n/2} (c x)^{n/2} \sqrt{\frac{b x^n}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a}}\right )}{\sqrt{b} c n \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(-1 + n/2)/Sqrt[a + b*x^n],x]

[Out]

(2*Sqrt[a]*(c*x)^(n/2)*Sqrt[1 + (b*x^n)/a]*ArcSinh[(Sqrt[b]*x^(n/2))/Sqrt[a]])/(Sqrt[b]*c*n*x^(n/2)*Sqrt[a + b
*x^n])

________________________________________________________________________________________

Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1+{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(-1+1/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^(-1+1/2*n)/(a+b*x^n)^(1/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1+1/2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate((c*x)^(1/2*n - 1)/sqrt(b*x^n + a), x)

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1+1/2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [A]  time = 13.4531, size = 31, normalized size = 0.57 \begin{align*} \frac{2 c^{\frac{n}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a}} \right )}}{\sqrt{b} c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(-1+1/2*n)/(a+b*x**n)**(1/2),x)

[Out]

2*c**(n/2)*asinh(sqrt(b)*x**(n/2)/sqrt(a))/(sqrt(b)*c*n)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1+1/2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate((c*x)^(1/2*n - 1)/sqrt(b*x^n + a), x)